Question

Find the coordinates of the center of curvature of the ellipse x²/a² + y²/b² = 1 or x = a cosθ, y = b sinθ. Hence show that the equation of its evolute is (ax) ^2/3 + (by^)2/3 = (a² – b²)^2/3

Answer 1

Given x = a cosθ, y = b sinθ                                                                                   ......(1)

and 

On taking cub of (6), squaring (7) and equating the two

Hence the locus of (x bar, y bar), i.e. the equation of evolute is 27ay² = 4(x – 2a)³